Teacher s Guide Geometry Semester A

Transcription

1 Teacher s Guide Geometry Semester A

2 Contents Overview... 3 Course Components... 4 Course Implementation Models... 8 Geometry A Overview... 9 Geometry A Curriculum Contents and Pacing Guide Unit 1: Introduction to Geometry and Transformations Unit 2: Congruence, Proof, and Constructions Unit 3: Similarity and Proof Unit 4: Trigonometry and Geometric Modeling EDMENTUM, INC.

3 Overview Plato Courses are developed to give the instructor a variety of ways to engage different learning modalities and to give the student an opportunity to experience a range of standards and objectives to ensure academic success. Plato Courses integrate Plato online curriculum, electronic Learning Activities, and supporting interactive activities. An array of assessment tools allows the instructor to correctly place students at the appropriate learning level, to evaluate strengths and needs, to create individualized learning goals, and to determine proficiency. Reports assist the student in understanding where he or she needs to focus to be academically successful as measured against objectives. Guidelines and tools are provided to track student progress and to determine a final course grade. Plato Courses give the instructor control over the instructional choices for individual students as well as for the classroom. The instructor may use all of the components as sequenced or select specific activities to support and enhance instruction. Plato Courses can be used in a variety of ways to increase student achievement. 3

4 Course Components Learning Activities Four types of learning activities form the building blocks of active learning for this course: lessons, activity-only lessons, Unit Activities, and Online Discussions. s. Each lesson in this course contains one or more learning components. All contain an interactive tutorial and a Activity. o Tutorials. Tutorials provide direct instruction and interactive checks of understanding. Practice interactions include drag-and-drop matching, multiplechoice questions, and fill-in-the-blank questions. Tutorials also often include links to informational websites, interactions, and videos, which enable students to broaden their understanding. o Activities. Activities are multipart problem-based activities that allow students to develop new learning in a constructivist way or to apply learning from the tutorial in a significant way. Activities are designed to be an authentic learning and assessment tool. Most Activities throughout the course employ a dynamic geometry tool, GeoGebra, helping learners to explore properties of geometric shapes and to test conjectures. Activity-Only s. Unlike traditional lessons, activity-only lessons are designed to address math standards that focus on narrow performance objectives. Students will demonstrate the ability to perform a specific task. Resembling regular Activities, activity-only lessons are multipart exercises and usually require students to use the geometry application GeoGebra. Unit Activities. The Unit Activity at the end of each unit aims to deepen understanding of key unit objectives and tie them together or tie them to other course concepts. Unit Activities are similar to Activities but are intended to combine and leverage concepts developed throughout the unit. Online Discussions. Online discussion with instructors and other students is a key activity, based on 21st-century skills, that allows for higher-order thinking about terminal objectives. An online threaded discussion mirrors the educational experience of a classroom discussion. Instructors can initiate a discussion by asking a complex, open-ended question. Students can engage in the discussion by responding both to the question and to the thoughts of others. Each unit in a course has one predefined discussion topic; instructors may include additional discussion topics. A rubric for grading discussion responses is included in this guide. 4

5 Learning Aids These learning aids assist students within the courseware activities: GeoGebra. GeoGebra is an interactive software application that allows learners to construct and modify geometric figures, take measurements, and describe geometric shapes algebraically. GeoGebra tools allow students to o create elements such as points, line segments, polygons, circles, and other curves either by direct placement in the graphics view or by inputting coordinates and equations; o set or define relationships between elements, such as defining a line parallel to another; o measure distances, angles, areas, and perimeters; and o determine the results of transformations. GeoGebra allows users to adjust the numerical accuracy of displayed measurements to a desired number of decimal places. However, in some cases, the values users see may not match the theoretical values precisely. In such cases, it is safe to assume a small margin of error due to rounding. Students can learn more about GeoGebra through online help from the manufacturer or the instructions contained in this user guide. Students will find links to both resources within the lessons. Scientific Calculator (Tutorials). The Scientific Calculator is available in case students do not have access to a handheld calculator. Data Tools (Probability and Statistics lessons only). Online data tools allow students to plot data using four widely accepted tools: histograms, box plots, scatter plots, and stem-and-leaf plots. Students can link to instructions for using these tools within the applicable lessons. Reader Support (Tutorials and Mastery Tests) enhances learning by enabling students to highlight text in the lesson and o play audio narration for the selected text (text to speech); o see a translation to another language for the selected text; and o see a dictionary definition in English or Spanish for a selected word. Assessment and Testing Best practices in assessment and testing call for a variety of activities to evaluate student learning. Multiple data points present a more accurate evaluation of student strengths and needs. These tools include both objective and authentic learning tools. 5

6 Objective Assessments. There is a specific learning objective associated with each lesson. Each lesson objective is assessed through objective assessments at three different points during the course: at the end of the specific lesson, at the end of the unit, and at the end of the semester. In addition, pretests based on these objectives are available at the beginning of each unit, if desired by the teacher. o Mastery tests at the end of each lesson provide the instructor and the student with clear indicators of areas of strength and weakness. These multiple-choice tests are taken online. o Unit pretests are optional assessments, typically designed for credit recovery use. If a student shows mastery of a lesson s objective (80% proficiency), the student may be automatically exempted from that lesson in the upcoming unit. Courses for first-time credit typically do not employ unit pretests. The tests are multiple-choice and are provided online. o Unit posttests help instructors track how well students have mastered the unit s content. The tests are multiple-choice and are provided online. o End-of-semester tests assess the major objectives covered in the course. By combining the unit pretest and unit posttest information with the end-ofsemester test results, the instructor will gain a clear picture of student progress. The tests are multiple-choice and are provided online. Authentic Learning Assessment. Of the assessment tools available in this course, three are designed specifically to address higher-level thinking skills and operations: Activities, Unit Activities, and Discussions. These authentic learning activities allow students to develop deep understanding and provide data for the teacher to assess knowledge development. These three types of activities are described in the Learning Activities section above. The following comments address their use for assessment. o Activities immerse the student into one or more in-depth problems that center on developing a deep understanding of the learning objective. They also provide a tool for assessing identified Common Core mathematical practices, inquiry skills, STEM skills, and 21st century skills. Most Activities in this course are self-checked by the student; however, it is possible to submit this work for teacher grading on paper, by , or by creating a drop box activity in the course learning path. In this course, activity-only lessons are teacher-graded Activities submitted through the drop box. These lessons allow the instructor to score work on a scale of 0 to 100. A 10-point suggested rubric is provided to both 6

7 the student and the teacher for this purpose. Teachers can find rubrics and sample answers for activity-only lessons in the Edmentum Support Center. o Unit Activities are similar to Activities, but are more time intensive and require a more integrative understanding of the unit s objectives. They also provide a tool for assessing identified Common Core mathematical practices, inquiry skills, STEM skills, and 21st century skills. Unit activities are teacher graded and are submitted through the drop box. These activities allow the instructor to score work on a scale of 0 to 100. A 10-point suggested rubric is provided to both the student and the teacher for this purpose. o Discussions encourage students to reflect on concepts, articulate their thoughts, and respond to the views of others. Thus, discussions help teachers assess students critical-thinking skills, communication skills, and overall facility with the unit concepts. Each unit in this course has one predefined discussion topic. Instructors can customize the course, however, to include additional discussion topics. Online discussions may use whatever rubric the instructor sets. A suggested rubric is provided here for reference. Relevance of Response Content of Response Participation D/F 0 69 Below Expectations The responses do not relate to the discussion topic or are inappropriate or irrelevant. Ideas are not presented in a coherent or logical manner. There are many grammar or spelling errors. The student does not make any effort to participate in the discussion. Online Discussion Rubric C Basic Some responses are not on topic or are too brief or low level. Responses may be of little value (e.g., yes or no answers). Presentation of ideas is unclear, with little evidence to back up ideas. There are grammar or spelling errors. The student participates in some discussions but not on a regular basis. B Proficient The responses are typically related to the topic and initiate further discussion. Ideas are presented coherently, although there is some lack of connection to the topic. There are few grammar or spelling errors. The student participates in most discussions on a regular basis but may require some prompting to post. A Outstanding The responses are consistently on topic and bring insight into the discussion, which initiates additional responses. Ideas are expressed clearly, with an obvious connection to the topic. There are rare instances of grammar or spelling errors. The student consistently participates in discussions on a regular basis. 7

8 Course Implementation Models Plato Courses give instructors the flexibility to define implementation approaches that address a variety of learning needs. Instructors can configure the courses to allow individual students to work at their own pace or for group or class learning. Furthermore, the courses can be delivered completely online (that is, using a virtual approach) or can include both face-to-face and online components (that is, using a blended approach). Depending on the learner grouping and learning approach, instructors can choose to take advantage of peer-to-peer interaction through Online Discussions. Similarly, if students have prior knowledge of the concepts taught in certain lessons, instructors can decide to employ unit pretests to assess students prior knowledge and exempt them from taking the lessons. Note, however, that this feature is primarily designed for credit recovery purposes. For first-time credit, students are typically not allowed to test out of course lessons. Following are two common implementation models for using Plato Courses, along with typical (but not definitive) implementation decisions. Independent Learning The student is taking the course online as a personal choice or as part of an alternative learning program. Learner grouping Learning approach Discussions Unit pretests independent learning blended or virtual remove from learning path students do not take pretests Group or Class Learning The online course is offered for a group of students. These students may not be able to schedule the specific course at their local school site, or they may simply want the experience of taking an online course. Learner grouping Learning approach Discussions Unit pretests group interaction blended or virtual use; additional discussion questions may be added students do not take pretests 8

9 Geometry A Overview Course Structure Geometry A is a one-semester course organized into units and lessons. The typical audience for this course is students at the high school level. Pedagogical Approach This course is designed to enable all students at the secondary level to develop a deep understanding of the geometry objectives identified in the course Pacing Guide detailed below. It is also based on the Common Core State Standards Initiative and on a modern understanding of student learning in mathematics and STEM disciplines. In addition to content standards, the Common Core State Standards Initiative makes these key points about CCSS high school mathematics curricula: They include rigorous content and application of knowledge through high-order skills. They call on students to practice applying mathematical ways of thinking to realworld issues and challenges, preparing students to think and reason mathematically. They set a rigorous definition of college and career readiness by helping students develop a depth of understanding and ability to apply mathematics to novel situations, as college students and employees regularly do. They emphasize mathematical modeling and the use of mathematics and statistics to analyze empirical situations, to understand them better, and to improve decisions. Activities and Unit Activities in this course exercise learning objectives that require ongoing attention throughout the student s education. These global learning objectives are specifically called out at the beginning of each Activity. They are organized into four useful categories: Mathematical Practices objectives address the eight Standards for Mathematical Practice identified by the Common Core State Standards Initiative. Inquiry objectives support skills associated with investigation, experimentation, analysis, drawing conclusions, and communicating effectively. STEM objectives stress activities that combine mathematics and other technical disciplines or that provide insight into careers in science, technology, engineering, and math. 9

10 21st Century Skills objectives call for using online tools, applying creativity and innovation, using critical-thinking and problem-solving skills, communicating effectively, assessing and validating information, performing large-scale data analysis, and carrying out technology-assisted modeling. Online Discussions require students to apply some similar skills to an interesting problem, with the added advantage that they enable communication and collaboration among students. This is a critical aspect of the course, especially in fully online implementations, where peer-to-peer interaction may be limited. Taken together, the elements of this course are designed to help students learn in a multifaceted but straightforward way. Finally, the curriculum is clearly relevant and highly engaging for students while being straightforward for teachers to manage. 10

11 Geometry A Curriculum Contents and Pacing Guide This course is divided into units and is designed to be completed in one semester. The Pacing Guide provides a general timeline for presenting each unit. This guide is adjustable to fit your class schedule. It is based on a typical 180-day school year schedule with 90 days per semester. Unit 1: Introduction to Geometry and Transformations Summary This unit focuses on a single CCSS domain that relates to basic geometric definitions and transformations in a plane: G.CO: Congruence Unit 1: Introduction to Geometry and Transformations Day Activity/Objective Common Core State Standard Type 1 day: 1 2 days: days: 4 5 Syllabus and Plato Student Orientation Review the Plato Student Orientation and Course Syllabus at the beginning of this course. Introduction to Geometry Become acquainted with the history, career applications, and logical structure and development of geometry. Basic Geometric Concepts Know precise definitions for the concepts of angle, circle, perpendicular line, parallel line, and line segment. G.CO.1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. Course Orientation 11

12 6 8 Representing Transformations in a Plane Represent transformations in a plane and compare transformations that preserve distance and angle to those that do not. G.CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch) Returning a Polygon to Its Original Position Describe the rotations and reflections that carry a given rectangle, parallelogram, trapezoid, or regular polygon onto itself. G.CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself Defining Rigid Transformations Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments Predicting Results of Rigid Transformations Predict the result of a rigid transformation and specify a sequence of transformations to carry a given figure onto another. G.CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. 4 days: Unit Activity and Discussion Unit 1 Includes applications or extensions of concepts from the following standard: G.CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). Unit Activity Discussion 1 day: 22 Posttest Unit 1 Assessment 12

13 Unit 2: Congruence, Proof, and Constructions Summary This unit focuses on a single CCSS domain that relates to congruence, proofs, and constructions: G.CO: Congruence Unit 2: Congruence, Proof, and Constructions Day Activity/Objective Common Core State Standard Type 2 days: days: days: Transformations and Congruence Use geometric descriptions of rigid motions to transform figures and use the definition of congruence in terms of rigid motions to decide if two figures are congruent. Sides and Angles of Congruent Triangles Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. ASA, SAS, and SSS Criteria for Congruent Triangles Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Lines, Angles, and Mathematical Proofs Learn to write mathematical proofs, and apply that knowledge to simple geometric relationships. Proving Theorems about Lines and Angles Prove theorems about lines and angles. G.CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. G.CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. G.CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints. s 13

14 34 36 Proving Theorems about Triangles Prove theorems about triangles. G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 ; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point Proving Theorems about Parallelograms Prove theorems about parallelograms. G.CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals Geometric Constructions with Lines and Angles Make formal geometric constructions with a variety of tools and methods. G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. 4 days: Unit Activity and Discussion Unit 2 Includes applications or extensions of concepts from the following standards: G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of in terms of rigid motions. G.CO.13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Unit Activity Discussion 1 day: 47 Posttest Unit 2 Assessment 14

15 Unit 3: Similarity and Proof Summary This unit focuses on a single CCSS domain that relates to similarity: G.SRT: Similarity, Right Triangles, and Trigonometry Unit 3: Similarity and Proof Day Activity/Objective Common Core State Standard Type days: Properties of Dilations Verify experimentally the properties of dilations given by a center and a scale factor. Similarity and Similarity Transformations Use the definition of similarity in terms of similarity transformations to decide whether two given figures are similar. AA, SAS, and SSS Criteria for Similar Triangles Use the properties of similarity transformations to establish the AA, SAS, and SSS criteria for two triangles to be similar. Similarity, Proportion, and Triangle Proofs Prove theorems about triangles using similarity relationships. G.SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor: A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. G.SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. 15

16 59 61 Using Congruence and Similarity with Triangles Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 4 days: Unit Activity and Discussion Unit 3 Includes applications or extensions of concepts from the following standards: Unit Activity Discussion G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. 1 day: 66 Posttest Unit 3 Assessment 16

17 Unit 4: Trigonometry and Geometric Modeling Summary This unit focuses on a single CCSS domain that relates to trigonometry and geometric modeling: G.SRT: Similarity, Right Triangles, and Trigonometry Unit 4: Trigonometry and Geometric Modeling Day Activity/Objective Common Core State Standard Type days: Trigonometric Ratios Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Sine and Cosine of Complementary Angles Explain and use the relationship between the sine and cosine of complementary angles. Solving Problems with Right Triangles Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Trigonometry and the Area of a Triangle Derive the formula A = 1/2 ab sin(c) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Proving the Laws of Sines and Cosines Prove the Laws of Sines and Cosines, and use them to solve problems. Applying the Laws of Sines and Cosines Understand and apply the Laws of Sines and Cosines to find unknown measurements in right and non-right triangles. G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles. G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. G.SRT.9 Derive the formula A = 1/2 ab sin(c) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. G.SRT.10 Prove the Laws of Sines and Cosines and use them to solve problems. G.SRT.11 Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and nonright triangles (e.g., surveying problems, resultant forces). 17

18 4 days: Unit Activity and Discussion Unit 4 Includes applications or extensions of concepts from the following standards: Unit Activity Discussion G.MG.1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). G.MG.3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. 1 day: 88 Posttest Unit 4 Assessment 1 day: 89 Semester Review 1 day: 90 End-of-Semester Test Assessment 18